Since the top of the sign has a length of 10 feet, split into two equal sections, then each section of the top is 5 feet.

The height of the triangle can be found by using the Pythagorean Theorem on either the left or right triangle.

Pythagorean Theorema² + b² = c² {sum of the squares of the legs is equal to the square of the hypotenuse}
5² + b² = 13² {substituted 5 in for a leg and 13 in for the hypotenuse, into the Pythagorean Theorem}
25 + b² = 169 {evaluated exponents}
b² = 144 {subtracted 25 from each side}
b = 12 {took square root of each side}

height of triangle is 12 ft

base x height
------------- = area of a triangle
2

10 x 12
------- = area {substituted 10 for base and 12 for height, of triangle, into area formula}
2